STABLE POISSON CONVERGENCE FOR INTEGER-VALUED RANDOM VARIABLES. Tsung-Lin Cheng* and Shun-Yi Yang 1. INTRODUCTION

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1 TAIWANESE JOURNAL OF MATHEMATICS Vol. 17, No. 6, , December 2013 DOI: /tjm This aer is available online at htt://journal.taiwanmathsoc.org.tw STABLE POISSON CONVERGENCE FOR INTEGER-VALUED RANDOM VARIABLES Tsung-Lin Cheng* and Shun-Yi Yang Abstract. In this aer, we obtain some stable Poisson Convergence Theorems for arrays of integer-valued deendent random variables. We rove that the limiting distribution is a mixture of Poisson distribution when the conditional second moments on a given σ-algebra of the sequence converge to some ositive random variable. Moreover, we aly the main results to the indicator functions of rowise interchangeable events and obtain some interesting stable Poisson convergence theorems. 1. INTRODUCTION It is likely that the greatest accomlishment of modern robability theory is the unified elegant theory of limits for sums of indeendent or stationary random variables. The former studies the limiting theorems when deendence structure is relaxed to comrising only indeendent random variables, while the later consider deendent but time-invariant distributed random variables. The mathematical theory of martingales may be regarded as an extension of the indeendence theory which has been firstly studied by Bernstein (1927) and Lévy (1935, 1937). Lévy introduced the conditional variance for martingales Vn 2 = E(Xj 2 F j 1 ), where (S n, F n,n 1) is a zero-mean, square integrable martingale and X n = S n S n 1 is the martingale difference. Doob (1953), Billingsley (1961), and Ibragimov (1963) established the Central Limit Theorem for martingales with stationary and ergodic differences. For such martingales the conditional variance is asymtotically constant; namely, (1) s 2 n Vn 2 Received March 6, 2013, acceted Aril 27, Communicated by Yuh-Jia Lee Mathematics Subject Classification: 60F05. Key words and hrases: Stable Poisson Convergence. *Corresonding author ,

2 1870 Tsung-Lin Cheng and Shun-Yi Yang where s 2 n = E(V n 2 ). Further extensions have been made by Rosén (1967a,b), Dvoretzky (1969, 1971, 1972) and Brown (1971), among others. Esecially, as indicated by Brown (1971), the crucial oint for martingale convergence is the condition (1) but stationarity or ergodicity. In McLeish (1974), an elegant roof about the martingale central limit theorem and invariance rinciles were given. The convergence of normalized martingales to more general distributions were investigated by Brown and Eagleson (1971), Eagleson (1976), Adler et al.(1978), among others. Since its first commencement in Rényi(1963), the concet of stable convergence has been largely extended and alied to many general setting and roblems concerning asymtotic behaviors of martingale arrays or stationary arrays. The stable convergence can be defined as follows. Let (Ω, F,P) be a robability sace, G be a sub-σ-algebra of F and (Y n ) d be a sequence of random variables. Let Y n Y. If for every B G,thereisa countable, dense set of oints x, such that lim P ({Y n x} B) =Q(x, B) n stably exists, then we say that Y n converges stably in G to Y and denote it by Y n F Y (or Y) on G, or (s,g) Y n Y. The maing Q(x, B) in the above definition is a distribution function when we fix B G, and it is a robability measure on G when we fix the real number x. If G = {, Ω}, then (s,g) d is just.ifg = F, then (s,g) stably is the usual Y n F Y (we dro the F), denoted by s. From the roof of Lemma 1 in Cheng and Chow (2002), we know that for almost every x in the value set of Y n, Q(x, ) is a measure absolutely continuous with resect to P. Unlike convergence in distribution, stable convergence in distribution is a roerty of sequences of random variables rather than that of their corresonding distribution functions. For more details about stable convergence we refer the readers to Aldous (1978). TheideabehindRényi s stable convergence is to generalize the renowned Central Limit Theorems to a mixture of normal distributions. The notion of a mixture of a well known distribution with a random arameter stems from Bayesian analysis. In Bayesian structure, a arameter emerging in the robability density function is assumed to be nonconstant, more generally, a random variable. The distribution of the random arameter is called the rior. Imagine a sequence of deendent random variables which converges in distribution to a well known distribution, say, to a normal distribution when conditioned on the event that the random arameter is fixed at some constant value. We will try to look for the osterior utilizing the observations at hand. Classical Poisson limit theorems assume the random variables to be i.i.d., integervalued or just be indeendent but not identically distributed. For an infinite exchangeable sequence of random variables, conditioned on the tail events, the sequence will

3 Stable Poisson Convergence for Integer-valued Random Variables 1871 behave like an i.i.d. sequence (Chow and Teicher 1997). However, this roerty doesn t hold for an array of finitely exchangeable random variables. Martingale methods rovide a unified aroach to both situations. The martingale method was suggested by Loynes (1969) in the context of U-statistics and develoed by Eagleson (1979, 1982) and Weber (1980) for exchangeable variables. A Poisson convergence theorem follows from the results for infinitely divisible laws develoed by Brown and Eagleson (1971). See also Freedman (1974). Theorem A. (Brown and Eagleson 1971, Freedman 1974). Let {A, F ; 1 k n, n 1} be an array of deendent events on the robability sace (Ω, F,P) (F F +1 F, A is F -measurable) and F n,0 be a sub-σ-algebra of F n,1. If λ>0 is a constant and for n, P(A F 1 ) max P(A F 1 ) 1 k n then S n = n I A d P oisson(λ). λ, 0, The conditions in Theorem A have also been used by Kalan(1977), Brown(1978) and Silverman and Brown(1978). However, so far as our knowledge goes, there were no literature discussing stable Poisson convergence. Therefore, we are going to try to derive a stable Poisson convergence theorem (SPCT, for abbreviation) with the limiting distribution of the tye of a Poisson mixture, namely, with the intensity arameter λ being a nonnegative random variable. In Cheng and Chow (2002), they roved an auxiliary lemma and obtained some interesting theorems on stable convergence to normal mixture. Under a mild (but not trivial) modification, we may obtain some theorems on stable Poisson convergence. This aer is organized as below: In Section 2, we consider λ to be a random variable and generalize the Poisson convergence theorem to comrises the stable Poisson convergence theorems by exloiting the conditional characteristic function introduced by Brown (1971), Hall and Heyde (1980), and Cheng and Chow (2002). In Section 3, we aly our main results to arrays of row-exchangeable events. Throughout this aer, all equalities and inequalities between random variables are in the sense of with robability one and I A denotes the indicator function of the set. All kinds of convergence, in distribution, in robability and in L, are denoted resectively by d,,and. L The fact that X n 0 in robability is abbreviated by X n = o (1). 2. MAIN RESULTS The following result is an auxiliary lemma for roving stable convergence.

4 1872 Tsung-Lin Cheng and Shun-Yi Yang Lemma 2.1. (Cheng and Chow, 2002). Y iff there exists a r.v. Y on (Ω I,G B,P u), wherei =(0, 1), B is the class of all Borel sets in I and u is the Lebesque measure on B, with the same distribution as Y such that for all real t, E(e ityn I A ) E(e ity I A I ), for all A G, and E(e ity I A I ) is continuous at t =0 for all A G. Y n (s,g) Let {A, F ; 1 k m n, n 1} be an array of events on the robability sace (Ω, F,P) and F n,0 be a sub-σ-algebra of F n,1.set X = I A, S n = m n X, and f n (t) = m n E 1(e it X ),wheree 1 (X)=E(X F 1 ). Theorem 2.1. Suose there exists a sub-σ-algebra G n=1 F n,0 and a ositive G-measurable random variable λ such that for n, (2) (3) (4) P(A F 1 ) max P(A F 1 ) 1 k m n E(e it S n f n (t) G) λ, 0, 0, then S n (s,g) Z, where the random variable Z has characteristic function E(ex{λ(e it 1)}) and for any A G,E(e it S n I A ) E(ex{λ(e it 1)}I A ). Proof. For clarity and convenience, we define E 1 (X) =E(X F 1 ). First, we want to show that f n (t) ex{λ(e it 1)}. According to the inequality e ix n (ix) k k=0 k! min{ 2 x n n!, x n+1 (n+1)! },wehave eix 1 ix min{2 x, x 2 2 }. We might set B(x) 2 x 2 and consider a function, say A(x), with A(x) B(x) such that e ix =1+ix + x A(x). It follows that E 1 (e it X ) = 1+iE 1 (t X )+te 1 ( X A(t X )) =1+itE 1 ( X )+ta(t)e 1 ( X ). Fix t, define δ ite 1 ( X )+ta(t)e 1 ( X ). Since A(t) 2, we have By (3), δ = ite 1 ( X )+ta(t)e 1 ( X ) 3 t E 1 ( X ). (5) max δ 3 t max E 1 ( X ) 0. 1 k m n 1 k m n

5 Stable Poisson Convergence for Integer-valued Random Variables 1873 Since for any z C with z 1, Hence, log(1 + z) z = z 0 1 ( 1+w 1)dw = z 0 w 1+w dw. log(1 + z) z z 0 w 1+w dw 1 z 1 z 0 w dw 1 1 z z 2 2, and log f n (t) = = log E 1 (e it X )= log(1 + δ ) m n δ + R n, where R n 1 2 δ 2 1 δ, and max 1 k mn δ < 1. On the set {w Ω;max 1 k mn δ < 1}, by(2),(3),wehave (6) R n 1 2 ( = δ 2 m 1 δ 1 n 1 max 1 k mn δ 1 max 1 k mn δ 1 max 1 k mn δ + max 1 k m n δ 1 max 1 k mn δ δ 2 ) δ 2 =(1+o (1)) δ 2 =9t 2 (1 + o (1)) E 2 1 ( X ) Ct 2 max E 1 ( X ) E 1 ( X ) 0. 1 k m n Hence, by (6), for all ε>0, asn, P( R n >ε)=p( R n >ε, max δ < 1) + P( R n >ε, max δ 1) 1 k m n 1 k m n P( R n >ε, max δ < 1) + P( max δ 1) 1 k m n 1 k m n 0.

6 1874 Tsung-Lin Cheng and Shun-Yi Yang Therefore, log f n (t) = δ + o (1) m n = it E 1 ( X )+ta(t) E 1 ( X )+o (1) itλ + ta(t)λ = λ(it + ta(t)) = λ(e it 1). Thus, f n (t) ex{λ(e it 1)}, and as a result, for all A G, By uniform integrability, f n (t) I A ex{λ(e it 1)} I A. E(f n (t) I A ) E[ex{λ(e it 1)} I A ]. By (4) and uniform integrability, for A G, Consequently, E[(e it S n f n (t)) I A ] 0. E(e it S n I A ) E(ex{λ(e it 1)}I A ). Next, ut β(ω, x)=p (X x λ), wherex is a mixture of a Poisson distribution with random intensity λ. DefineY (ω, y) on (Ω I,G B,P u) as in Lemma 1 of Cheng and Chow (2002), i.e. Y (ω, y)=inf{x : <x<,β(ω, x) y}. Then ) 1 E (e ity (ω,y) I A I = dp e ity (ω,y)dy A 0 dp e itx dβ(ω, x)= ex{λ(e it 1)}dP A = E[ex{λ(e itx 1)} I A ]. By Lemma 2.1, we comlete the roof. Remark 2.1. It seems that the conditions of Theorem 2.1 are stronger than Theorem A. It is because that, similar to Theorem 1 in Cheng and Chow (2002), the stable convergence is a generalization of the classical distributional convergence. When G = {, Ω}, condition (4) is satisfied automatically (see Remark 2.2 below), and in this case, Theorem A is valid as a result. The function f n (t) lays an imortant role in roving the stable convergence. However, when conditioned on G {, Ω}, itmight not be coincident with E(e it S n ) which is crucial in roving distributional convergence. It will be interesting to study the conditions imlying the coincidence of f n (t) and E(e it S n ) when conditioned on G {, Ω}. A

7 Stable Poisson Convergence for Integer-valued Random Variables 1875 In order to obtain a more comlete version of SPCT, we add a condition to the original assumtions of Theorem 2.1 to ensure the SPCT for the artial sum S n. Let {X, F ;1 k m n } be any array of nonnegative integer-valued random varibles on (Ω, F,P) and F n,0 be a sub-σ-algebra of F n,1. Set S n = mn X, X = I {X =1}, S n = mn X,andf n (t) = mn E 1 (e it X ). Corollary 2.1. Let λ be a ositive G-measurable random variable, where G is a sub-σ-algebra of F. Ifforn, (7) (8) P(X =1 F 1 ) max P(X =1 F 1 ) 1 k m n λ, 0, then f n (t) ex{λ(e it 1)}. Moreover, if G n=1 F n,0 and (9) (10) E(e it S n f n (t) G) 0, P(X 2) 0, then S n (s,g) Z, where where the random variable Z has characteristic function E(ex{λ(e it 1)}) and for A G,E(e itsn I A ) E(ex{λ(e it 1)}I A ). Proof. Let A = {X =1}. From (10), we have P(S n S n ) = P(S n S n > 0) m n P(X 2) 0. Therefore, by Theorem 2.1, we have comleted the roof. Remark 2.2. When λ is a constant, according to the Theorem 3 of Beśka(1982), we have S d n P oisson(λ), iffn (t) ex{λ(e it d 1)}. And by (10), S n P oisson(λ). From Corollary 2.1 and Remark 2.2, we can obtain the following corollary concerning classical Poisson convergence theorem for arrays of indeendent integer-valued random variables. Corollary 2.2. Let X, 1 k m n, be indeendent nonnegative integer-valued

8 1876 Tsung-Lin Cheng and Shun-Yi Yang random variables. If λ>0 is a constant and for n, then P(X =1) λ, max P(X =1) 0, 1 k m n P(X 2) 0, S n = X d P oisson(λ). The following theorem can be roven in a fashion similar to Theorem 2.1 given in Section 2. Theorem 2.2. Suose there exists a sub-σ-algebra G n=1 F n,0 such that {A ; 1 k m n, n 1} be conditional indeendent given G in each row. Let λ be a ositive G-measurable random variable. If for n, (11) (12) P(A G) max P(A G) 1 k m n λ, 0, then S n (s,g) Z, where the random variable Z has characteristic function E(ex{λ(e it 1)}) and for A G,E(e it S n I A ) E(ex{λ(e it 1)}I A ). Proof. By the same way of Theorem 2.1, we have m n which imlies that for any A G, E(e it X G) ex{λ(e it 1)}, m n E(e it X G) I A ex{λ(e it 1)} I A. Due to the roerty of conditional indeendence, E(e it S m n n G)= G)= E(e it X G).

9 Stable Poisson Convergence for Integer-valued Random Variables 1877 Hence, (e it S n I A G) ex{λ(e it 1)} I A. Since {E(e it S n G); n 1} is uniformly integrable, for A G E(e it Sn I A ) E(ex{λ(e it 1)} I A ). 3. APPLICATIONS Let {A,, 2,...,m n,n 1} be an array of row-exchangeable events. Set S n,j = j I A, F n,0 = σ( m n I A ),andf n,j = σ(i An,1,I An,2,...,I An,j, mn k=j+1 I A ),forj =1, 2,...,m n. Let f n (t) = n E 1(e iti A ). Similar to Eagleson (1979), we may obtain the following results. Theorem 3.3. Let G be a sub-σ-algebra of n=1 F n,0, and λ a ositive G- measurable random variable. If for n, (13) (14) (15) (16) n P(A n,1 G) L 1 λ, n 2 P(A n,1 G) 2 n 2 P(A n,1 A n,2 G) L 1 0, n P(A n,1 A c n,2 ) 0, m n /n, then S n,n (s,g) Z, where the random variable Z has characteristic function E(ex{λ(e it 1)}) and for A G,E(e itsn,n I A ) E(ex{λ(e it 1)}I A ). (17) Proof. By (13), we have np(a n,1 ) E(λ) < which yields (18) P(A n,1 ) 0. Similarly, we have which imlies E[n 2 P(A n,1 G) 2 n 2 P(A n,1 A n,2 G)] 0, (19) n 2 E(I An,1 P(A n,1 G)) n 2 P(A n,1 A n,2 ) 0. Fix n, j, 1 j m n.

10 1878 Tsung-Lin Cheng and Shun-Yi Yang By exchangeability, for any B F n,j 1, and each k, j k m n, (20) E(I An,j I B )=E(I A I B ). Hence, for any B F n,j 1, mn k=j E(I An,j I B )=E( I A m n j +1 I B). And mn k=j I A m n j+1 is F n,j 1 -measurable, we have E(I An,j F n,j 1 )= mn k=j I A m n j +1. Similarly, for each t with j t m n, (21) P(A n,t F n,j 1 )= mn k=j I A m n j +1. In articular, for each j with 1 j m n, (22) P(A n,j F n,j 1 )=P(A n,mn F n,j 1 ). Moreover, for each t with 1 t m n, (23) P(A n,1 G)=E(P(A n,j G) F n,0 )=E(P(A n,1 F n,0 ) G) = E(P(A n,t F n,0 ) G)=P(A n,t G). Now, we want to claim that max 1 j n P(A n,j F n,j 1 ) 0. Since for each n 1, {P(A n,mn F n,j 1 ), F n,j 1, 1 j n} is a martingale. Hence, by (21), (22) and Doob s inequality, for any ε>0, P( max 1 j n P(A n,j F n,j 1 ) >ε)=p( max 1 j n P(A n,m n F n,j 1 ) >ε) E[P(A n,m n F n,n 1 )] ε = P(A n,m n ) ε = P(A n,1) ε 0 Next, claim that n P(A n,j F n,j 1 ) λ. By (13), we only need to show that n P(A n,j F n,j 1 ) n P(A n,1 G) 0.

11 Stable Poisson Convergence for Integer-valued Random Variables 1879 Since for each j with 1 j n, by (13), (14), (15) and Jensen s inequality, we have (E P(A n,j F n,j 1 ) P(A n,1 G) ) 2 =(E P(A n,mn F n,j 1 ) P(A n,mn G) ) 2 E P(A n,mn F n,j 1 ) P(A n,mn G) 2 = E(I An,mn P(A n,mn F n,j 1 )) E(I An,mn P(A n,mn G)) m n I A k=j = E(I An,mn m n j +1 ) E(I A n,mn P(A n,mn G)) P(A n,1) m n j +1 + P(A n,1 A n,2 ) E(I An,mn P(A n,mn G)), which imlies E P(A n,j F n,j 1 ) P(A n,1 G) [ P(A 1 n,1) m ] 2 n j+1 +[P(A n,1 A n,2 ) E(I An,mn P(A n,mn G))] 1 2. Moreover, by (13)-(16), we have E P(A n,j F n,j 1 ) P(A n,1 G) [n P(A n,1 )] 1 1 n 2 [ m n n ] 2 +[n 2 P(A n,1 A n,2 ) n 2 E(I An,mn P(A n,mn G))] 1 2 E(λ) 0+0=0. Hence, for any ε>0, P( P(A n,j F n,j 1 ) n P(A n,1 G) >ε) 1 ε E [P(A n,j F n,j 1 ) P(A n,1 G)] 1 ε E P(A n,j F n,j 1 ) P(A n,1 G) 0. Therefore, n P(A n,j F n,j 1 ) λ. Next, since for any fixed n, 1 k n,

12 1880 Tsung-Lin Cheng and Shun-Yi Yang I A E(I A F 1 ) = I A = = 1 m n k +1 1 m n k +1 j=k+1 mn 1 m n k +1 [ j=k+1 we have for n, by (15), I An,j I A + j=k E I A E(I A F 1 ) =2 1 m n k +1 I An,j I A c j=k (I A I An,j I A + I An,j I A c ) (I A I A c n,j + I An,j I A c )], m n k m n k+1 P(A n,1a c n,2 ) 2 P(A n,1 A c n,2 )=2n P(A n,1a c n,2 ) 0. Note that for any fixed n, 1 k n, I A c E 1 (I A c ) = 1 I A 1+E 1 (I A ) = E 1 (I A ) I A = I A E 1 (I A ). Therefore we have for n, = E( e iti A E 1 (e iti A ) ) E e it I A + I A c e it E 1 (I A ) E 1 (I A c ) E I A E 1 (I A ) + So, for n, E I A c E 1 (I A c ) 0. P{ E(e itsn,n f n (t) G) >ε} P{E( e itsn,n f n (t) G) >ε} 1 ε E( eitsn,n f n (t) ) 1 ε E( e iti A E 1 (e iti A ) ) 0. By Theorem 2.1, we have comleted the roof.

13 Stable Poisson Convergence for Integer-valued Random Variables 1881 Let {A } k 1 be an array of rowise exchangeable events. Set S n,j = j I A, G n,j = σ( j I A,I An,j+1,I An,j+2,...),j 1, and G n, = G n,j, G = n=1 G n,. Similar to Weber(1980), we may obtain the following results. Theorem 3.4. If there exists a G-measurable integrable random variable λ such that for n, (24) (25) n P(A n,1 G n, ) λ, n P(A n,1 A c n,2) 0, then S n,n (s,g) Z, where the random variable Z has characteristic function E(ex{λ(e it 1)}) and for A G,E(e itsn,n I A ) E(ex{λ(e it 1)}I A ). Proof. Fix n, j. By exchangeability, for any B G n,j, and each k, 1 k j, E(I An,1 I B )=E(I A I B ). Hence, for any B G n,j, j E(I An,1 I B )=E( I A j I B ). And j I A j is G n,j -measurable, we have (26) j P(A n,1 G n,j )= A. j For each n, since G n,j G n,j+1, then {P(A n,1 G n,j ), G n,j,n 1} is a reversed martingale, and by the reversed martingale convergence theorem, as j, P(A n,1 G n,j ) L 2 (27) P(A n,1 G n, ). Next, for some m N, we consider σ-field m F n,j 1 σ(i An,1,I An,2,...,I An,j 1, I A ), then for fixed n and j, by(26) and (27), as m, m k=j P(A n,j F n,j 1 )= I A m j +1 = ( m I A j 1 I A ) m j +1 m = m j +1 P(A n,1 G n,m ) j 1 m j +1 P(A n,1 G n,j 1 ) L 1 P(An,1 G n, ), k=j

14 1882 Tsung-Lin Cheng and Shun-Yi Yang which in turn imlies for each n, asm, P(A n,j F n,j 1 ) L 1 n P(A n,1 G n, ). Thus, for each n, there exists m n N such that for any m m n n, wehave E( P(A n,j F n,j 1 ) n P(A n,1 G n, ) ) < 1 n. Hence, for each n, we can choose σ-fields F n,j 1 σ(i A n,1,i An,2,...,I An,j 1, mn k=j I A ), 1 j n, such that So, by (24), P(A n,j Fn,j 1) n P(A n,1 G n, ) L 1 0. P(A n,j Fn,j 1) λ. And, in the same way as Theorem 3.1, we have max 1 j n P(A n,j Fn,j 1 ) 0. Taking G = n=1 F n,0 and f n(t) = n E(eitI A F 1 ), we also have E(e itsn,n f n (t) G) 0. Then, by Theorem 2.1, we comlete the roof. REFERENCES 1. R. J. Adler and D. J. Scott, Martingale central limit theorems without negligibility conditions, Corrigendum. Bull. Austral. Math. Soc., 18 (1978), D. J. Aldous and G. K. Eagleson, On mixing and stability of limit theorems, Ann. Probab., 6 (1978), D. J. Aldous, Exchangeability and Related Toics, Lecture Notes in Mathematics, Sringer-Verlag, A. D. Barbour and L. Holst, Some alications of the Stein-Chen method for roving Poisson convergence, Adv. Al. Prob., 21 (1989), S. Bernstein, Sur l extension du théoréme limite du calcul des robabilitiés aux sommes de quantités déendantes, Math. Ann., 85 (1927), 1-59.

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16 1884 Tsung-Lin Cheng and Shun-Yi Yang 25. I. S. Helland, Central limit theorems for martingales with discrete or continuous time, Scand. J. Statist, 9 (1982), I. A. Ibragimov, A central limit theorem for a class of deendent random variables, Theory Probab. Al., 8 (1963), N. Kalan, Two alications of a Poisson aroximation for deendent events, Ann. Prob., 5 (1977), R. G. Laha and V. K. Rohagi, Probability Theory, P. Lévy, Proriétés asymtotiques des sommes de variables aléatoires enchainées, Bull. Sci. Math., 59(ser.2) (1935a), 84-96, P. Lévy, Proriétés asymtotiques des sommes de variables aléatoires indeendantes ou enchainées, J. Math. Pures Al., bf 14(ser.9) (1935b), P. Lévy, Théorie de l adition des Variables Aléatoires, Gauthier-Villars, Paris, R. M. Loynes, The central limit theorem for backwards martingales, Z. Wahrsch. Verw. Gebiete., 13 (1969), D. L. McLeish, Deendent central limit theorem and invariance rinciles, Ann. Probab., 2 (1974), A. Rényi, On stable sequences of events, Sankhya Ser. A, 25 (1963), A. Rényi, Foundations of Probability, Holden-Day, San Francisco, B. Rosén, On the central limit theorem for sums of deendent random variables, Z. Wahrsch. Verw. Gebiete., 7 (1967a), Rosén, B. On asymtotic normality of sums of deendent random variables, Z. Wahrsch. Verw. Gebiete, 7 (1967b), A. N. Shiryayev, Probability, Sringer, Berlin, New York, A. G. Sholomitskii, On the necessary conditions of normal convergence for martingales, Theory Probab. Al., 43 (1998), A. G. Sholomitskii, On the necessary conditions of oisson convergence for martingales, Theory Probab. Al., 49 (2005), B. W. Silverman and T. C. Brown, Short distances, flat triangles and Poisson limits, J. Al. Prob., 15 (1978), N. C. Weber, A Martingale Aroach to Central Limit Theorems for Exchangeable Random Variables, J. Al. Prob., 17 (1980), J. Wesollowski, Poisson Process via Martingale and Related Characteristics, J. Al. Prob., 36 (1999),

17 Stable Poisson Convergence for Integer-valued Random Variables 1885 Tsung-Lin Cheng and Shun-Yi Yang Deartment of Mathematics National Changhua University of Education Changhua 500, Taiwan